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Faster computation of Bernoulli numbers. (English) Zbl 0755.11006
The author presents an algorithm, based on the classical formula $$B\sb{2k}=(-1)\sp{k+1}2(2k!)\zeta(2k)(2\pi)\sp{-2k},$$ to compute the $2k$th Bernoulli number $B\sb{2k}$, defined by $X/(e\sp X- 1)=\sum\sb{n\geq 1}B\sb n X\sp n/n!$. The space requirement of this algorithm is ${\cal O}(n\log\log n)$ bits and it involves ${\cal O}(n\sp 2\log\sp 2n\log\log n)$ bit operations. The algorithm can also be efficiently generalized to compute all Bernoulli numbers up to some point. This slightly improves on the previous algorithms of {\it S. Chowla} and {\it P. Hartung} [Acta Arith. 22, 113--115 (1972; Zbl 0244.10008)] and of {\it D. E. Knuth} and {\it T. J. Buckholtz} [Math. Comput. 21, 663--688 (1967; Zbl 0178.04401)].

##### MSC:
 11Y16 Algorithms; complexity (number theory) 11B68 Bernoulli and Euler numbers and polynomials 68W30 Symbolic computation and algebraic computation
##### Keywords:
algorithm; Bernoulli numbers
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